Functional Differential Equations With Delay and Random Effects

The authors study the existence of mild solutions to a functional differential equation with delay and random effects. They use a random fixed point theorem with a stochastic domain. An example is included to illustrate their results.     

A new method for numerical differentiation based on direct and inverse problems of partial differential equations

In this paper, we mainly study a numerical differentiation problem which aims to approximate the second order derivative of a single variable function from its noise data. By transforming the problem into a combination of direct and inverse problems of partial differential equations (heat conduction equations), a new method that we call the PDEs-based numerical differentiation method is proposed. By means of the finite element method and the Tikhonov regularization, implementations of the proposed PDEs-based method are presented with a posterior strategy for choosing regularization parameters. Numerical results show that the PDEs-based numerical differentiation method is highly feasible and stable with respect to data noise.     

Uncertain multiobjective traveling salesman problem

Traveling salesman problem is a fundamental combinatorial optimization model studied in the operations research community for nearly half a century, yet there is surprisingly little literature that addresses uncertainty and multiple objectives in it. A novel TSP variation, called uncertain multiobjective TSP (UMTSP) with uncertain variables on the arc, is proposed in this paper on the basis of uncertainty theory, and a new solution approach named uncertain approach is applied to obtain Pareto efficient route in UMTSP. Considering the uncertain and combinatorial nature of UMTSP, a new ABC algorithm inserted with reverse operator, crossover operator and mutation operator is designed to this problem, which outperforms other algorithms through the performance comparison on three benchmark TSPs. Finally, a new benchmark UMTSP case study is presented to illustrate the construction and solution of UMTSP, which shows that the optimal route in deterministic TSP can be a poor route in UMTSP.     

Interpolating minimal energy C1‐Surfaces on Powell–Sabin Triangulations: Application to the resolution of elliptic problems

In this article, we present a method to obtain a C¹‐surface, defined on a bounded polygonal domain Ω, which interpolates a specific dataset and minimizes a certain “energy functional.” The minimization space chosen is the one associated to the Powell–Sabin finite element, whose elements are C¹‐quadratic splines. We develop a general theoretical framework for that, and we consider two main applications of the theory. For both of them, we give convergence results, and we present some numerical and graphical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 798–821, 2015     

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015     

On independent sets in random graphs

The independence number of a sparse random graph G(n,m) of average degree d = 2m/n is well‐known to be with high probability, with in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size , i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size for any fixed (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point . Roughly speaking, we prove that independent sets of size form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 436–486, 2015     

One-dimensional migration of interstitial clusters in SUS316L and its model alloys at elevated temperatures

For self-interstitial atom (SIA) clusters in various concentrated alloys, one-dimensional (1D) migration is induced by electron irradiation around 300 K. But at elevated temperatures, the 1D migration frequency decreases to less than one-tenth of that around 300 K in iron-based bcc alloys. In this study, we examined mechanisms of 1D migration at elevated temperatures using in situ observation of SUS316L and its model alloys with high-voltage electron microscopy. First, for elevated temperatures, we examined the effects of annealing and short-term electron irradiation of SIA clusters on their subsequent 1D migration. In annealed SUS316L, 1D migration was suppressed and then recovered by prolonged irradiation at 300 K. In high-purity model alloy Fe-18Cr-13Ni, annealing or irradiation had no effect. Addition of carbon or oxygen to the model alloy suppressed 1D migration after annealing. Manganese and silicon did not suppress 1D migration after annealing but after short-term electron irradiation. The suppression was attributable to the pinning of SIA clusters by segregated solute elements, and the recovery was to the dissolution of the segregation by interatomic mixing under electron irradiation. Next, we examined 1D migration of SIA clusters in SUS316L under continuous electron irradiation at elevated temperatures. The 1D migration frequency at 673 K was proportional to the irradiation intensity. It was as high as half of that at 300 K. We proposed that 1D migration is controlled by the competition of two effects: induction of 1D migration by interatomic mixing and suppression by solute segregation.     

Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation

In nonlinear physics,it is very difficult to study interactions among different types of nonlinear waves.In this paper,the nonlocal symmetry related to the truncated Painleve′expansion of the(2+1)-dimensional Burgers equation is localized after introducing multiple new variables to extend the original equation into a new system.Then the corresponding group invariant solutions are found,from which interaction solutions among different types of nonlinear waves can be found.Furthermore,the Burgers equation is also studied by using the generalized tanh expansion method and a new Ba¨cklund transformation(BT)is obtained.From this BT,novel interactive solutions among different nonlinear excitations are found.